Faucet problem with Body Force.

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QUESTION
 
Ferrofluids are fluids that respond to magnetic fields. So, if you impose a magnetic field on a ferrofluid,
you can make the body force be whatever magnitude and direction you like, depending on how the
magnetic field is applied.
Suppose that you have the “faucet” problem that we did (are doing?) for homework. Now, however, you
have a body force that is gravity plus an additional downward force that changes with vertical distance.
Suppose you have the following body force
Fbody = ⇢g ↵zk
Now, assume that you are given
• the flow rate Q
• the diameter d0 at “section 0”
• the length L
• the value of potential function .
Use the combination of (a) the macroscopic mass balance on water, and (b) Bernoulli’s principle to
predict the diameter, d1 and the velocity, v1 of the stream at the downstream (“section 1”) end with the
information given? You should be able to generate an explicit formula; please do not put any numerical
values in at this juncture. To be extra clear, here are the equations I would like you to start with the
equations listed below.
Conservation of mass
Z
V
@⇢
@t +
Z
A
⇢v · n = 0
Bernoulli’s principle
1
2 ⇢v2
0 + (z0) + p0 = 1
2 ⇢v2
1 + (z1) + p1
Here, for Bernoulli’s principle, you need to use a more general potential energy. Instead of = ⇢gz, you
will have to use
(z) = Z z
0
⇢g + ↵z k dz
 

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